# Connections between sequential Bayesian inference and evolutionary dynamics

arXiv: https://arxiv.org/abs/2411.16366 · HTML: https://mlsanity.com/paper/2411.16366/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 86 references that are in the map.
- First posted on arXiv 25 Nov 2024.

## Builds on (most cited first, 16 of 16)

- [Accelerating Langevin Sampling with Birth-death](https://mlsanity.com/paper/1905.09863/) (2019)
- [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022)
- [Sampling in Unit Time with Kernel Fisher-Rao Flow](https://mlsanity.com/paper/2401.03892/) (2024)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [Riemannian game dynamics](https://mlsanity.com/paper/1603.09173/) (2016)
- [A connection between Tempering and Entropic Mirror Descent](https://mlsanity.com/paper/2310.11914/) (2023)
- [Tikhonov Regularization Within Ensemble Kalman Inversion](https://mlsanity.com/paper/1901.10382/) (2019)
- [Measure transport with kernel mean embeddings](https://mlsanity.com/paper/2401.12967/) (2024)
- [Poisson's equation in nonlinear filtering](https://mlsanity.com/paper/1412.5845/) (2014)
- [Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2406.17263/) (2024)
- [McKean-Vlasov SDEs in nonlinear filtering](https://mlsanity.com/paper/2007.12658/) (2020)
- [Kernel Approximation of Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2410.20622/) (2024)
- [On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering](https://mlsanity.com/paper/2006.08843/) (2020)
- [Kalman Filter and its Modern Extensions for the Continuous-time Nonlinear Filtering Problem](https://mlsanity.com/paper/1702.07241/) (2017)
- [On the stability of matrix-valued Riccati diffusions](https://mlsanity.com/paper/1808.00235/) (2018)
- [Lie algebra structure of fitness and replicator control](https://mlsanity.com/paper/2005.09792/) (2020)

## Built on by (most cited first, 4 of 4)

- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2506.05905/) (2025)
- [An operator splitting analysis of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2511.18060/) (2025)
- [Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2609.18118/) (2026)

## Abstract (arXiv)

It has long been posited that there is a connection between the dynamical equations describing evolutionary processes in biology and sequential Bayesian learning methods. This manuscript describes new research in which this precise connection is rigorously established in the continuous time setting. Here we focus on a partial differential equation known as the Kushner-Stratonovich equation describing the evolution of the posterior density in time. Of particular importance is a piecewise smooth approximation of the observation path from which the discrete time filtering equations, which are shown to converge to a Stratonovich interpretation of the Kushner-Stratonovich equation. This smooth formulation will then be used to draw precise connections between nonlinear stochastic filtering and replicator-mutator dynamics. Additionally, gradient flow formulations will be investigated as well as a form of replicator-mutator dynamics which is shown to be beneficial for the misspecified model filtering problem. It is hoped this work will spur further research into exchanges between sequential learning and evolutionary biology and to inspire new algorithms in filtering and sampling.